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\author{学号 \underline{\hspace{4cm}} \hspace{1cm} 姓名 \underline{\hspace{4cm}} }
\title{实变函数练习3.3 - 可测集类 }
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\date{2024 年 4 月 15 日}
%\date{March 9, 2021}

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\begin{document}

\maketitle

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\begin{enumerate}

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\item  %Problem 01
设 $E\subseteq \mathbb{R}^n$ 是一个点集。证明下述结论： 
\begin{enumerate}
\item  若 $m^*(E)=0$, 则 $E$ 是勒贝格可测集。
\item  若 $m^*(E)=0$, 设 $F\subseteq E$, 则 $E$ 是勒贝格可测集。
\item  可数个零测度集的并集仍是零测度集。
\end{enumerate}




\vspace{0.2cm}

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\item  %Problem 02
证明 $\mathbb{R}^n$ 中的开区间、闭区间和半开半闭区间都是勒贝格可测集。

\vspace{0.2cm}

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\item  %Problem 03
证明 $\mathbb{R}^n$ 中的开集和闭集都是勒贝格可测集。

\vspace{0.2cm}

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\item  %Problem 04
证明 $\mathbb{R}^n$ 中的博雷尔集都是勒贝格可测集。

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\item  %Problem 05
设 $E \subseteq \mathbb{R}^n$ 是一个勒贝格可测集。证明存在一列开集的交集 $G$ 使得 
$E \subseteq G$ 且 $m(G-E)=0$. 

\vspace{0.2cm}

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\item  %Problem 06
设 $E \subseteq \mathbb{R}^n$ 是一个勒贝格可测集。证明 
\begin{eqnarray*}
m(E) &=& \inf \,\{ \, m(G): G \text{是开集, } E\subseteq G \} \\ 
&=& \sup \,\{ \, m(K): K \text{是紧集, } K\subseteq E \}.
\end{eqnarray*}

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\item  %Problem 07
%H11.
设 $\{E_n\mid n=1,2,\cdots\}$ 是一列勒贝格可测集。
证明下极限 $\varliminf\limits_{n\to\infty} E_n$ 也是勒贝格可测集，且有
$$m\left( \varliminf_{n\to\infty} E_n \right) \le \varliminf_{n\to\infty} m(E_n). $$


\vspace{0.2cm}

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\item  %Problem 08
%H13.
设 $E\subseteq [0,1]$ 是一个勒贝格可测集，且 $m(E)=1$. 设 $A\subseteq [0,1]$ 也是勒贝格可测集。
证明 $$m(E\cap A) = m(A).$$

\vspace{0.2cm}


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\item  %Problem 09
%H16.
设 $E\subseteq \mathbb{R}$ 是一个勒贝格可测集，设 $c\in\mathbb{R}$. 记 $cE=\{cx: x\in E\}$. 
证明 $cE$ 也是勒贝格可测集。

\vspace{0.2cm}

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\item  %Problem 10
举例说明直线上存在勒贝格不可测集。

\vspace{0.2cm}


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\end{enumerate}


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\end{document}

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